Point (1,0) is closest to which of the following lines? A.

Soln. I too agree with answer A. Drawing the quadrants and trying to plot the lines approximately i suppose will help to solve this problem.

my approach is just visualize each line in mind
( i know its easier said than done...but considering 2 mins time....i think this is the fastest way)
refer image attached.......Now the task is to find min distance from a line in options to the point (1,0).......this can also be done by a formula from a geometry chapter

First visualise point (1,0)...now to calculate min distance....here its very helpful to know a property.....in 45-45-90 triangle.....sides are in 1:1:srt(2) proportion....now apply this in following options

1) y=x....since y=x.....aangle of line wrto x axis = 45 ( shown in fig)....we need perpendicular distance hence another angle = 90 deg....thus it becomes a triangle of 45-45-90....hence hypoteneous is 1...now equate the proportion as 1:1:srt(2) = x:x:1.....this gives x = 1/srt(2)min distance = 1/srt(2)

2)y=1....min distance = 1
3)y+x=3......min distance = srt(2).......apply the same logic as of option 1
4)x=2.....min distance = 1
5)x+y=-1....min distance = srt(2).......apply the same logic as of option 1


OA A

Best approach is to sketch the lines.
Practice getting the y-intercept and the x-intercepts.
These two intercepts will guide you win in sketching your lines.

1) (0,0) (1,1) (2,2) ... (n,n)
2) draw a line that intersects y=1 (This is quickly beaten by y=x. OUT!)
3) (0,3) (3,0) (This is quickly beaten by y=x. OUT!)
4) draw a line that intersects x=2
5) (0,-1) (-1,0) (This is quickly beaten by y=x. OUT!)

Looking at the scetch you will notice that (4) x=2 has a minimum distance of 1.
Looking at the scetch of (1) x=y has a minimum distance of d < 1

Look at the diagram below:



As you can see point (1,0) is closest to line y = x.

Answer: A.

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